Some Invariants of Cartesian Product of a Path and a Complete Bipartite Graph

Ramy Shaheen; Suhail Mahfud; Qays Alhawat

doi:doi:10.11648/j.dmath.20190402.11

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Some Invariants of Cartesian Product of a Path and a Complete Bipartite Graph

Ramy Shaheen, Suhail Mahfud, Qays Alhawat

Published in International Journal of Discrete Mathematics (Volume 4, Issue 2)

Received: 11 November 2019 Accepted: 16 December 2019 Published: 6 January 2020

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Abstract

A topological index of graph G is a numerical parameter related to G, which characterizes its topology and is preserved under isomorphism of graphs. Properties of the chemical compounds and topological indices are correlated. In this paper we will compute M-polynomial, first and second Zagreb polynomials and forgotten polynomial for the Cartesian Product of a path and a complete bipartite graph for all values of n and m. From the M-polynomial, we will compute many degree-based topological indices such that general Randić index, inverse Randić index, first and second Zagreb index, modified Zagreb index, Symmetric division index, Inverse sum index augmented Zagreb index and harmonic index for the Cartesian Product of a path and a complete bipartite graph. Also, we will compute the hyper- Zagreb index, the first and second multiple Zagreb index and forgotten index for the Cartesian Product of a path and a complete bipartite graph.

Published in	International Journal of Discrete Mathematics (Volume 4, Issue 2)
DOI	10.11648/j.dmath.20190402.11
Page(s)	61-70
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

M-polynomial, Topological Index, Path, Complete Graph, Cartesian Product

References

[1]	H. Hosoya, On some counting polynomials in chemistry, Discrete Appl. Math. 19 (1988) 239–257.
[2]	E. Deutsch, S. Klavžar, Computing Hosoya polynomials of graphs from primary subgraphs, MATCH Commun. Math. Comput. Chem. 70 (2013) 627–644.
[3]	M. Eliasi, A. Iranmanesh, Hosoya polynomial of hierarchical product of graphs, MATCH Commun. Math. Comput. Chem. 69 (2013) 111–119.
[4]	X. Lin, S. J. Xu, Y. N. Yeh, Hosoya polynomials of circumcoronene series, MATCH Commun. Math. Comput. Chem. 69 (2013) 755–763.
[5]	I. Gutman, A formula for the Wiener number of trees and its extension to graphs containing cycles, Graph Theory Notes New York. 27 (1994) 9-15.
[6]	H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc. 69 (1947) 17-20.
[7]	G. G. Cash, Relationship between the Hosoya polynomial and the hyper-Wiener index, Appl. Math. Lett. 15 (2002) 893–895.
[8]	F. M. Brückler, T. Doŝlić, A. Graovac, I. Gutman, On a class of distance-based molecular structure descriptors, Chem. Phys. Lett. 503 (2011) 336–338.
[9]	E. Deutsch, S. Klavžar. M-Polynomial, and degree-based topological indices. Iran. J. Math. Chem. 6 (2015) 93–102.
[10]	M. Munir, W. Nazeer, S. Rafique and S. Kang, M. M-polynomial and related topological indices of Nanostar dendrimers. Symmetry. 8 (2016) 97. doi: 10.3390/sym8090097.
[11]	M. Munir, W. Nazeer, Z. Shahzadi and S. M. Kang, Some invariants of circulant graphs. Symmetry 8 (2016) 134. doi: 10.3390/sym8110134.
[12]	I. Gutman and N. Trinajstic, Graph theory and molecular orbitals total φ-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972) 535–538.
[13]	K. Das and I. Gutman, Some properties of the second Zagreb index, MATCH Commun. Math. Comput. Chem. 52 (2004) 103–112.
[14]	I. Gutman and K. C. Das, The first Zagreb indices 30 years after, MATCH Commun. Math. Comput. Chem. 50 (2004) 83–92.
[15]	S. Nikolić, G. Kovačević, A. Miličević and N. Trinajstić, The Zagreb indices 30 years after, Croat. Chem. Acta. 76 (2003) 113–124.
[16]	N. Trinajstic, S. Nikolic, A. Milicevic and I. Gutman, On Zagreb indices, Kem. Ind. 59 (2010) 577–589.
[17]	D. Vukičević and A. Graovac, Valence connectivity versus Randić, Zagreb and modified Zagreb index: A linear algorithm to check discriminative properties of indices in acyclic molecular graphs, Croat. Chem. Acta. 77 (2004) 501–508.
[18]	A. Milicevic, S. Nikolic and N. Trinajstic, On reformulated Zagreb indices, Mol. Divers. 8 (2004) 393–399.
[19]	M. Randić, On the characterization of molecular branching, J. Amer. Chem. Soc. 97 (1975) 6609–6615.
[20]	B. Bollobas and P. Erdös, Graphs of extremal weights, Ars Combin. 50 (1998) 225–233.
[21]	D. Amic, D. Beslo, B. Lucic, S. Nikolic and N. Trinajstić, The vertex-connectivity index revisited, J. Chem. Inf. Comput. Sci. 38 (1998) 819–822.
[22]	Y. Hu, X. Li, Y. Shi, T. Xu and I. Gutman, On molecular graphs with smallest and greatest zeroth- Corder general Randić index, MATCH Commun. Math. Comput. Chem. 54 (2005) 425–434.
[23]	B. Furtula, A. Graovac and D. Vukičević, Augmented Zagreb index, J. Math. Chem. 48 (2010) 370–380.
[24]	Y. Huang, B. Liu and L. Gan. Augmented Zagreb Index of Connected Graphs. MATCH Commun. Math. Comput. Chem. 67 (2012) 483–494.
[25]	V. K. Gupta, V. Lokesha, S. B. Shwetha and P. S. Ranjini, On the symmetric division deg index of graph, Southeast Asian Bull. Math. 40 (2016) 59–80
[26]	A. T. Balaban, Highly discriminating distance based numerical descriptor, Chem. Phys. Lett. 89 (1982) 399–404.
[27]	S. Fajtlowicz, On conjectures of Graffiti II, Congr. Numer. 60 (1987) 189–197.
[28]	O. Favaron, M. Mahéo and J. F. Saclé, Some eigenvalue properties in graphs (conjectures of Graffiti-II), Discrete Math. 111 (1993) 197–220.
[29]	G. H. Shirdel, H. R. Pour and A. M. Sayadi, The hyper-Zagreb index of graph operations. Iran. J. Math. Chem. 4 (2013) 213–220.
[30]	M. Ghorbani and N. Azimi, Note on multiple Zagreb indices. Iran. J. Math. Chem. 3 (2012) 137–143.
[31]	R. Shaheen, S. Mahfud and Q. Alhawat, Some topological indices and polynomials of Cartesian Product of two Paths. Submitted to Bulletin of Computational Mathematics (2019).

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APA Style

Ramy Shaheen, Suhail Mahfud, Qays Alhawat. (2020). Some Invariants of Cartesian Product of a Path and a Complete Bipartite Graph. International Journal of Discrete Mathematics, 4(2), 61-70. https://doi.org/10.11648/j.dmath.20190402.11

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ACS Style

Ramy Shaheen; Suhail Mahfud; Qays Alhawat. Some Invariants of Cartesian Product of a Path and a Complete Bipartite Graph. Int. J. Discrete Math. 2020, 4(2), 61-70. doi: 10.11648/j.dmath.20190402.11

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AMA Style

Ramy Shaheen, Suhail Mahfud, Qays Alhawat. Some Invariants of Cartesian Product of a Path and a Complete Bipartite Graph. Int J Discrete Math. 2020;4(2):61-70. doi: 10.11648/j.dmath.20190402.11

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@article{10.11648/j.dmath.20190402.11,
  author = {Ramy Shaheen and Suhail Mahfud and Qays Alhawat},
  title = {Some Invariants of Cartesian Product of a Path and a Complete Bipartite Graph},
  journal = {International Journal of Discrete Mathematics},
  volume = {4},
  number = {2},
  pages = {61-70},
  doi = {10.11648/j.dmath.20190402.11},
  url = {https://doi.org/10.11648/j.dmath.20190402.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.dmath.20190402.11},
  abstract = {A topological index of graph G is a numerical parameter related to G, which characterizes its topology and is preserved under isomorphism of graphs. Properties of the chemical compounds and topological indices are correlated. In this paper we will compute M-polynomial, first and second Zagreb polynomials and forgotten polynomial for the Cartesian Product of a path and a complete bipartite graph for all values of n and m. From the M-polynomial, we will compute many degree-based topological indices such that general Randić index, inverse Randić index, first and second Zagreb index, modified Zagreb index, Symmetric division index, Inverse sum index augmented Zagreb index and harmonic index for the Cartesian Product of a path and a complete bipartite graph. Also, we will compute the hyper- Zagreb index, the first and second multiple Zagreb index and forgotten index for the Cartesian Product of a path and a complete bipartite graph.},
 year = {2020}
}

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TY  - JOUR
T1  - Some Invariants of Cartesian Product of a Path and a Complete Bipartite Graph
AU  - Ramy Shaheen
AU  - Suhail Mahfud
AU  - Qays Alhawat
Y1  - 2020/01/06
PY  - 2020
N1  - https://doi.org/10.11648/j.dmath.20190402.11
DO  - 10.11648/j.dmath.20190402.11
T2  - International Journal of Discrete Mathematics
JF  - International Journal of Discrete Mathematics
JO  - International Journal of Discrete Mathematics
SP  - 61
EP  - 70
PB  - Science Publishing Group
SN  - 2578-9252
UR  - https://doi.org/10.11648/j.dmath.20190402.11
AB  - A topological index of graph G is a numerical parameter related to G, which characterizes its topology and is preserved under isomorphism of graphs. Properties of the chemical compounds and topological indices are correlated. In this paper we will compute M-polynomial, first and second Zagreb polynomials and forgotten polynomial for the Cartesian Product of a path and a complete bipartite graph for all values of n and m. From the M-polynomial, we will compute many degree-based topological indices such that general Randić index, inverse Randić index, first and second Zagreb index, modified Zagreb index, Symmetric division index, Inverse sum index augmented Zagreb index and harmonic index for the Cartesian Product of a path and a complete bipartite graph. Also, we will compute the hyper- Zagreb index, the first and second multiple Zagreb index and forgotten index for the Cartesian Product of a path and a complete bipartite graph.
VL  - 4
IS  - 2
ER  -

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Author Information

Ramy Shaheen

Department of Mathematics, Faculty of Science Tishreen University, Lattakia, Syria
Suhail Mahfud

Department of Mathematics, Faculty of Science Tishreen University, Lattakia, Syria
Qays Alhawat

Department of Mathematics, Faculty of Science Tishreen University, Lattakia, Syria

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Plain Text BibTeX RIS

APA Style

Ramy Shaheen, Suhail Mahfud, Qays Alhawat. (2020). Some Invariants of Cartesian Product of a Path and a Complete Bipartite Graph. International Journal of Discrete Mathematics, 4(2), 61-70. https://doi.org/10.11648/j.dmath.20190402.11

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ACS Style

Ramy Shaheen; Suhail Mahfud; Qays Alhawat. Some Invariants of Cartesian Product of a Path and a Complete Bipartite Graph. Int. J. Discrete Math. 2020, 4(2), 61-70. doi: 10.11648/j.dmath.20190402.11

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AMA Style

Ramy Shaheen, Suhail Mahfud, Qays Alhawat. Some Invariants of Cartesian Product of a Path and a Complete Bipartite Graph. Int J Discrete Math. 2020;4(2):61-70. doi: 10.11648/j.dmath.20190402.11

Copy | Download

@article{10.11648/j.dmath.20190402.11,
  author = {Ramy Shaheen and Suhail Mahfud and Qays Alhawat},
  title = {Some Invariants of Cartesian Product of a Path and a Complete Bipartite Graph},
  journal = {International Journal of Discrete Mathematics},
  volume = {4},
  number = {2},
  pages = {61-70},
  doi = {10.11648/j.dmath.20190402.11},
  url = {https://doi.org/10.11648/j.dmath.20190402.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.dmath.20190402.11},
  abstract = {A topological index of graph G is a numerical parameter related to G, which characterizes its topology and is preserved under isomorphism of graphs. Properties of the chemical compounds and topological indices are correlated. In this paper we will compute M-polynomial, first and second Zagreb polynomials and forgotten polynomial for the Cartesian Product of a path and a complete bipartite graph for all values of n and m. From the M-polynomial, we will compute many degree-based topological indices such that general Randić index, inverse Randić index, first and second Zagreb index, modified Zagreb index, Symmetric division index, Inverse sum index augmented Zagreb index and harmonic index for the Cartesian Product of a path and a complete bipartite graph. Also, we will compute the hyper- Zagreb index, the first and second multiple Zagreb index and forgotten index for the Cartesian Product of a path and a complete bipartite graph.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - Some Invariants of Cartesian Product of a Path and a Complete Bipartite Graph
AU  - Ramy Shaheen
AU  - Suhail Mahfud
AU  - Qays Alhawat
Y1  - 2020/01/06
PY  - 2020
N1  - https://doi.org/10.11648/j.dmath.20190402.11
DO  - 10.11648/j.dmath.20190402.11
T2  - International Journal of Discrete Mathematics
JF  - International Journal of Discrete Mathematics
JO  - International Journal of Discrete Mathematics
SP  - 61
EP  - 70
PB  - Science Publishing Group
SN  - 2578-9252
UR  - https://doi.org/10.11648/j.dmath.20190402.11
AB  - A topological index of graph G is a numerical parameter related to G, which characterizes its topology and is preserved under isomorphism of graphs. Properties of the chemical compounds and topological indices are correlated. In this paper we will compute M-polynomial, first and second Zagreb polynomials and forgotten polynomial for the Cartesian Product of a path and a complete bipartite graph for all values of n and m. From the M-polynomial, we will compute many degree-based topological indices such that general Randić index, inverse Randić index, first and second Zagreb index, modified Zagreb index, Symmetric division index, Inverse sum index augmented Zagreb index and harmonic index for the Cartesian Product of a path and a complete bipartite graph. Also, we will compute the hyper- Zagreb index, the first and second multiple Zagreb index and forgotten index for the Cartesian Product of a path and a complete bipartite graph.
VL  - 4
IS  - 2
ER  -

Copy | Download